Infinitely many non-radial solutions to a critical Choquard equation

2025/07/21 by Caputo, Sabrina, Vaira, Giusi
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.15747

Abstract

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation -Δu +V(|x|) u =(|x|^-μ* |u|^{2^⋆_μ})|u|^{2^⋆_μ-2}u,\quadin \mathbb R^N where V(|x|) is a bounded, nonnegative and symmetric potential in \mathbb R^N with N≥ 5, 0<μ≤ 4, * stands for the standard convolution and 2^⋆_μ:=(2N-μ)/(N-2) is the upper critical exponent in the sense of the Hardy - Littlewood - Sobolev inequality. By applying a finite dimensional reduction method we prove that if r²V(r) has a local maximum point or local minimum point r₀>0 with V(r₀)>0 then the problem has infinitely many non-radial solutions with arbitrary large energies.

Citations

• Infinitely many solutions for a boundary Yamabe problem
• Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations
• Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities
• Uniqueness and nondegeneracy of solutions for a critical nonlocal equation
• Semiclassical states for Choquard type equations with critical growth: critical frequency case
• Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well
• Uniqueness of positive solutions with Concentration for the Schrödinger-Newton problem
• Singularly perturbed critical Choquard equations
• Nondegeneracy of ground states and multiple semiclassical solutions of the Hartree equation for general dimensions
• Nodal solutions for the Choquard equation
• Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics
• Classification of solutions for an integral equation
• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• Electrons in lattice fields

Related

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• On the critical Choquard equation with potential well
• Nonlinear Choquard equations: doubly critical case
• A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality
• Normalized solutions to lower critical Choquard equation in mass-supercritical setting